ECE 6332, Spring, 2016
Wireless Communication
Zhu Han
Department of Electrical and Computer Engineering
Class 13
Mar. 7th, 2016
Outline

Section 5.5
– Inter-Symbol-Interference
– Nyguist Three Criteria
– Pulse Shaping
– Partial Response
ISI Example
sequence sent
1
sequence received 1
0
1
1(!)
1
Signal received
Threshold
t
0
-3T
-2T
-T
0
T
2T
3T
4T
Sequence of three pulses (1, 0, 1)
sent at a rate 1/T
5T
Baseband binary data transmission system.

ISI arises when the channel is dispersive

Frequency limited -> time unlimited -> ISI

Time limited -> bandwidth unlimited -> bandpass channel ->
time unlimited -> ISI
p(t)
ISI


First term : contribution of the i-th transmitted bit.
Second term : ISI – residual effect of all other transmitted bits.

We wish to design transmit and receiver filters to minimize the
ISI.

When the signal-to-noise ratio is high, as is the case in a
telephone system, the operation of the system is largely limited
by ISI rather than noise.
ISI

Nyquist three criteria
– Pulse amplitudes can be
detected correctly despite pulse
spreading or overlapping, if
there is no ISI at the decisionmaking instants



1: At sampling points, no
ISI
2: At threshold, no ISI
3: Areas within symbol
period is zero, then no ISI
– At least 14 points in the finals


4 point for questions
10 point like the homework
1st Nyquist Criterion: Time domain
p(t): impulse response of a transmission system (infinite length)
p(t)
1
 shaping function
0
no ISI !
t
1
T
2 fN
t0
Equally spaced zeros,
-1
interval
1
T
2 fn
2t0
1st Nyquist Criterion: Time domain
Suppose 1/T is the sample rate
The necessary and sufficient condition for p(t) to satisfy
1, n  0
pnT   
0, n  0
Is that its Fourier transform P(f) satisfy

 P f  m T   T
m  
1st Nyquist Criterion: Frequency domain

 P f  m T   T
m  
0
fa  2 f N
f
4 fN
(limited bandwidth)
Proof
Fourier Transform

pt    P f exp  j 2ft df


pnT    P f exp  j 2fnT df
At t=T


pnT  
 2 m 1 2T
 
m  



m  
1 2T

1 2T
1 2T

1 2T
1 2T
1 2T
2 m 1 2T
P f  exp  j 2fnT df
P f  m T  exp  j 2fnT df

 P f  m T exp  j 2fnT df
m  
B f  exp  j 2fnT df
B f  

 P f  m T 
m  
Proof
B f  

 P f  m T 

B f  
b
n
n  
m  
bn  T 
1 2T
1 2T
bn  Tp nT 
B f   T
exp  j 2nfT 
T
bn  
0

B  f exp  j 2nfT 
n  0
n  0
 P f  m T   T
m  
Sample rate vs. bandwidth

W is the bandwidth of P(f)

When 1/T > 2W, no function to satisfy Nyquist condition.
P(f)
Sample rate vs. bandwidth
When 1/T = 2W, rectangular function satisfy Nyquist
condition

T ,  f  W 
sin t T
 t 
pt  
 sinc   P f   
,
t
T 
0, otherwise 
1
0.8
Spectra
0.6
0.4
0.2
0
-0.2
-0.4
0
1
2
3
4
Subcarrier Number k
5
6
Sample rate vs. bandwidth

When 1/T < 2W, numbers of choices to satisfy Nyquist
condition

A typical one is the raised cosine function
Cosine rolloff/Raised cosine filter

Slightly notation different from the book. But it is the same
sin(  Tt ) cos( r Tt )
prc0 (t ) 

t
T
1  (2 r Tt ) 2
r : rolloff factor
0  r 1
1
Prc0 ( j 2f ) 
1
2
1  cos(
0
f
(
2 r T  r  1))


f  (1  r ) 21T
if
1
2T
(1  r )  f 
f 
1
2T
(1  r )
1
2T
(1  r )
Raised cosine shaping

Tradeoff: higher r, higher bandwidth, but smoother in time.
W
P(ω)
r=0
r = 0.25
r = 0.50
r = 0.75
r = 1.00

0
π
W

0
ECE 4371 Fall 2008
ω
2w
W
p(t)
π
W
t
Figure 4.10 Responses for different rolloff factors.
(a) Frequency response. (b) Time response.
Cosine rolloff filter: Bandwidth efficiency

Vestigial spectrum
data rate
1/ T
2 bit/s
 rc 


bandwidth (1  r ) / 2T 1  r Hz
bit/s
1
Hz

2
(1  r )
bit/s
 2
Hz


2nd Nyquist (r=1)
r=0
2nd Nyquist Criterion

Values at the pulse edge are distortionless

p(t) =0.5, when t= -T/2 or T/2; p(t)=0, when t=(2k-1)T/2, k≠0,1
-1/T ≤ f ≤ 1/T
Pr ( f )  Re[
PI ( f )  Im[

 (1)
n
P ( f  n / T )]  T cos( fT / 2)
n
P ( f  n / T )]  0
n  

 (1)
n  
Example
3rd Nyquist Criterion

Within each symbol period, the integration of signal (area) is
proportional to the integration of the transmit signal (area)

 ( wt ) / 2
,w

 sin( wT / 2)
T
P ( w)  

 0,
w 

T

1
p(t ) 
2
2 n1T
2
A  2 n1
2
T
 /T
( wt / 2)
jwt
e
dw

sin( wT / 2)
 / T
1,
p(t )dt  
0,
n0
n0
Eye Diagram

Eye diagram is a means of evaluating the quality of a received
“digital waveform”
– By quality is meant the ability to correctly recover symbols and
timing
– The received signal could be examined at the input to a digital
receiver or at some stage within the receiver before the decision
stage





Eye diagrams reveal the impact of ISI and noise
Two major issues are 1) sample value variation, and 2) jitter and
sensitivity of sampling instant
Eye diagram reveals issues of both
Eye diagram can also give an estimate of achievable BER
Check eye diagrams at the end of class for participation
Interpretation of Eye Diagram
Raised Cosine Eye Diagram

The larger , the wider the
opening.

The larger , the larger
bandwidth (1+ )/Tb

But smaller  will lead to larger
errors if not sampled at the best
sampling time which occurs at
the center of the eye.
Cosine rolloff filter: Eye pattern
2nd Nyquist
1st Nyquist:
1st Nyquist:
2nd Nyquist:
2nd Nyquist:
1st Nyquist
1st Nyquist:
2nd Nyquist:

2nd Nyquist:
1st Nyquist:
Eye Diagram Setup

Eye diagram is a retrace display of
data waveform
– Data waveform is applied to
input channel
– Scope is triggered by data
clock
– Horizontal span is set to cover
2-3 symbol intervals

Measurement of eye opening is
performed to estimate BER
– BER is reduced because of
additive interference and noise
– Sampling also impacted by
jitter
Partial Response Signals

Previous classes: Sy(w)=|P(w)|^2 Sx(w)
– Control signal generation methods to reduce Sx(w)
– Raise Cosine function for better |P(w)|^2

This class: improve the bandwidth efficiency
– Widen the pulse, the smaller the bandwidth.
– But there is ISI. For binary case with two symbols, there is only
few possible interference patterns.
– By adding ISI in a controlled manner, it is possible to achieve a
signaling rate equal to the Nyquist rate (2W symbols/sec) in a
channel of bandwidth W Hertz.
Example

Duobinary Pulse
– p(nTb)=1, n=0,1
– p(nTb)=1, otherwise

Interpretation of received signal
– 2: 11
– -2: 00
– 0: 01 or 10 depends on the previous transmission
Duobinary signaling

Duobinary signaling (class I partial response)
Duobinary signal and Nyguist Criteria

Nyguist second criteria: but twice the bandwidth
Differential Coding

The response of a pulse is spread over more than one signaling
interval.

The response is partial in any signaling interval.

Detection :
– Major drawback : error propagation.

To avoid error propagation, need deferential coding (precoding).
Modified duobinary signaling

Modified duobinary signaling
– In duobinary signaling, H(f) is nonzero at the origin.
– We can correct this deficiency by using the class IV partial
response.
Modified duobinary signaling

Spectrum
Modified duobinary signaling

Time Sequency: interpretation of receiving 2, 0, and -2?
Pulse Generation

Generalized form of
correlative-level
coding
(partial response signaling)
Tradeoffs

Binary data transmission over a physical baseband channel can
be accomplished at a rate close to the Nyquist rate, using
realizable filters with gradual cutoff characteristics.

Different spectral shapes can be produced, appropriate for the
application at hand.

However, these desirable characteristics are achieved at a price :
– A large SNR is required to yield the same average probability of
symbol error in the presence of noise.
Other types of partial response signals
paper
Type
r0
r1
r2
r3
r4
ideal
1
I
1
1
II
1
2
1
5
III
2
1
-1
6
IV
1
0
-1
3
V
-1
0
2
p(t)
P(W)
Levels
2
3
0
-1
5